The Arc Length of a Smooth, Planar Curve and Distance Traveled
Connect functional arc length to the coordinate-speed formula, distinguishing curve length from area.
Before this connection: Functional-curve arc length; Parametric coordinate rates.
College Board Topic 8.13 · CED p. 159 (PDF p. 164) · CED effective Fall 2020; project course outline 2026–27.
BC extension: a short curve segment is approximately a right-triangle hypotenuse, not a rectangle of area. For y=f(x) with continuous f′ on [a,b], length is L=∫ₐᵇ√(1+[f′(x)]²)dx. The 1 represents horizontal motion, and the squared derivative converts it into vertical motion. A curve moving downward still adds positive length. The units are length, not square units.
For x=g(y), use L=∫꜀ᵈ√(1+[g′(y)]²)dy with y-bounds. Check smoothness before invoking the formula on a whole interval; split at corners and add the lengths of smooth pieces. Some length integrals have no convenient elementary antiderivative, so a correct setup plus permitted numerical integration is useful. Arc length is the distance along the graph. It differs from area under the graph, and a horizontal width alone usually undercounts it. Motion along more general parametrized curves is developed in BC Unit 9; this topic handles curves given as functions.
Let's work through it
- For y=(2/3)x^(3/2) on [0,3], f′=√x. Reason: Differentiate the smooth function before forming the length integrand.
- L=∫₀³√(1+x)dx. Reason: Square f′ inside 1+[f′]².
- Evaluate (2/3)[(1+x)^(3/2)]₀³=(2/3)(8−1)=14/3. Reason: A simple substitution evaluates the length integral.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Does a negative f′ make negative length?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Does a negative f′ make negative length?
Worked answer and reason
No.
The derivative is squared and the length integrand is nonnegative.
Retrieval 2
How does a straight line’s arc length compare with endpoint distance?
Worked answer and reason
They agree.
The formula reduces to the Pythagorean distance for a constant slope.
Catch the tempting mistake
- Arc length is ∫|f(x)|dx.
- That gives area between the graph and the axis. Length uses √(1+[f′]²), built from small horizontal and vertical distances.
Calculator workspace and limits
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Put it to work
- Unit 8 problem 25 · No calculator · analytic
- Unit 8 problem 26 · No calculator · analytic
- Unit 8 problem 27 · Graphing calculator required · analytic
- Unit 8 problem 28 · Graphing calculator required · analytic
- Unit 8 problem 29 · No calculator · verbal
- Unit 8 problem 30 · No calculator · multiple
For the teacher
- Derive the formula from a tiny right triangle before using it. Compare area, horizontal distance, and arc length for one line. Keep shared AB applications active: this extension adds length, rather than replacing area, volume, and net change.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.